The function you have provided is:

$f(x) = \frac{8x}{x^2 + 4}$

Key aspects of this function:

1. Domain: The function is defined for all real values of $x$ because the denominator $x^2 + 4$ never equals zero (since $x^2$ is always non-negative and 4 is positive).

2. Simplification: The function does not simplify further because the numerator and denominator have no common factors.

3. Horizontal Asymptote: As $x \to \infty$ or $x \to -\infty$, the degree of the denominator (2) is greater than the degree of the numerator (1). Therefore, the horizontal asymptote is at $y = 0$.

4. Vertical Asymptote: There are no vertical asymptotes because the denominator $x^2 + 4$ does not have real roots (it cannot be zero for real values of $x$).

5. Critical Points: To find critical points, you can take the derivative of the function and set it to zero to find where the slope is zero.

Would you like me to go into detail on any of these points? Here are some related questions to think about:

1. How do we find the derivative of this function?
2. What is the behavior of this function at $x = 0$?
3. How do we sketch this function using critical points and asymptotes?
4. What are the inflection points of this function?
5. How can we analyze the end behavior of the function more rigorously?

Tip: When finding horizontal asymptotes for rational functions, always compare the degree of the numerator and the denominator to get a sense of the function's long-term behavior.